**1. Introduction**

At first-order phase transitions, coexisting macroscopic domains of different phases emerge from small fluctuations of a homogeneous phase. Late stages of this process are often dominated by the motion of the interfaces separating the domains. Considering this large-time coarsening behavior, i.e., the growth of a characteristic length scale *l*(*t*) as *t* → ∞ determines important characteristics of the dynamics and led to the identification of several universality classes of domain growth. We are here concerned with conserved order parameters, for which the expected behavior is *l*(*t*) ∼ *t*1/3, which results from the scale invariance of the Mullins–Sekerka system *x* → *λ<sup>x</sup>*, *t* → *λ*3*t*. Rigorous results exist for an upper bound for *l*(*t*), stating that microstructures cannot coarsen faster than the similarity rate [1]. As there are non-generic configurations, e.g., stripe domains with zero curvature which are stable, lower bounds cannot be expected within a deterministic framework. Besides this scaling law, solutions with random initial data are also believed to be statistically self-similar in this large-time regime. Numerical studies based on a Cahn–Hilliard equation and related coarse-grained theories indicate that the approach of the large-time regime with the statistically self-similar structures might be very slow [2]. To explore these regimes numerically thus requires large length and time scales, which limits the accessible sample size. We are here interested in these statistically self-similar structures, which have been used for various art and design projects, e.g., [3]. Here, we would like to explore very large, in principle infinite, samples. To tackle such a system we consider, instead of one huge simulation, many moderately sized domains with different initial data, and require the boundaries to match. If appropriately done, this will allow construction of large (infinite) tilings which are statistically self-similar. With a random arrangemen<sup>t</sup> of finitely many computed structures, the impression of an infinite tiling with no recurrence could be achieved. For this impression, the boundary conditions at the computational domains are crucial. They are described in detail in Section 4 together with the finite-element approach to solve the Cahn–Hilliard equation. In Section 2 we show various results, among other things a computer program which allows navigation through space and time of an infinitely extended structure. We further discuss improvements and outline possible applications. In Section 3, we discuss scaling and self-similar properties.
